Learn Portuguese (Portugal): Quero comprar uma relíquia genuína o mais barato possível. - I want to buy a genuine relic as cheaply as possible.

In this sentence, barato is acting as an adverb describing how you want to buy it ("cheaply"), rather than an adjective describing the relic itself. Because it functions as an adverbial phrase, both the article o and the word barato remain in their default, invariable masculine singular form. If you wanted to describe the relic itself as "the cheapest relic possible", it would change to match the noun: a relíquia mais barata possível.

If it means "cheaply", why don't we use an adverb ending in "-mente" like baratamente?

While baratamente is technically a valid word, everyday Portuguese strongly prefers using the masculine singular adjectives barato (cheap/cheaply) and caro (expensive/expensively) as adverbs when talking about buying, selling, or costing. Saying o mais baratamente possível would sound quite unnatural to a native speaker.

Could I move o mais barato possível to a different part of the sentence?

It is best kept exactly where it is, at the end of the sentence. Because it modifies the verb comprar (to buy), it typically follows the direct object (uma relíquia genuína). Squeezing it between the verb and the object (Quero comprar o mais barato possível uma relíquia...) sounds clunky and breaks the natural flow of the sentence.

Just as a refresher, why does come after here? Could it go before?